Solve (5sqrt{3}-sqrt{5})^2-sqrt{7^2}*sqrt{15}

$$( 5 \sqrt { 3 } - \sqrt { 5 } ) ^ { 2 } - \sqrt { 7 ^ { 2 } } \cdot \sqrt { 15 }$$

Evaluate

$$80-17\sqrt{15}\approx 14.159283114$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5\sqrt{3}-\sqrt{5}\right)^{2}.

$$25\left(\sqrt{3}\right)^{2}-10\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\sqrt{7^{2}}\sqrt{15}$$

The square of \sqrt{3} is 3.

$$25\times 3-10\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\sqrt{7^{2}}\sqrt{15}$$

Multiply 25 and 3 to get 75.

$$75-10\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\sqrt{7^{2}}\sqrt{15}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$75-10\sqrt{15}+\left(\sqrt{5}\right)^{2}-\sqrt{7^{2}}\sqrt{15}$$

The square of \sqrt{5} is 5.

$$75-10\sqrt{15}+5-\sqrt{7^{2}}\sqrt{15}$$

Add 75 and 5 to get 80.

$$80-10\sqrt{15}-\sqrt{7^{2}}\sqrt{15}$$

Calculate 7 to the power of 2 and get 49.

$$80-10\sqrt{15}-\sqrt{49}\sqrt{15}$$

Calculate the square root of 49 and get 7.

$$80-10\sqrt{15}-7\sqrt{15}$$

Combine -10\sqrt{15} and -7\sqrt{15} to get -17\sqrt{15}.

$$80-17\sqrt{15}$$